Is 509 a Prime Number?

There are two types of numbers, mostly —

Prime numbers and composite numbers, depending on the number of factors.

A prime number is a natural number that is divisible only by 1 and itself.

For example, 3 is a prime number because it is divisible by 1 and itself.

A composite number is a positive number that is divisible by more than two numbers.

For example, 6 is divisible by 1, 2, 3, and 6 making it a composite number.

Prime numbers follow few properties like

• Prime numbers are positive numbers always greater than 1.
   
• 2 is the only even prime number.
   
• They have only two factors: 1 and the number itself.
   
• Any two distinct prime numbers are co-prime numbers because they have only one common factor that is 1.

As 509 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 509 has exactly two factors, it is a prime number. Few methods are used to distinguish between prime and composite numbers. A few methods are:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization

The method in which we count the number of divisors to categorize the numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize prime and composite numbers. If there is a total count of only 2 divisors, then the number would be prime. If the count is more than 2, then the number is composite. Let’s check whether 509 is prime or composite.

Step 1: All numbers are divisible by 1 and itself.

Step 2: Divide 509 by 2. It is not divisible by 2, so 2 is not a factor of 509.

Step 3: Divide 509 by 3. It is not divisible by 3, so 3 is not a factor of 509.

Step 4: You can simplify checking divisors up to 509 by finding the square root value, which is approximately 22.5. We then need to only check divisors up to this value.

Step 5: As we continue to check divisors up to 22, we find that 509 is not divisible by any number except 1 and 509 itself.

Since 509 has only 2 divisors, it is a prime number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 9, which is odd, so 509 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 509 is 14. Since 14 is not divisible by 3, 509 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 9. Therefore, 509 is not divisible by 5.

Divisibility by 7: Performing the divisibility test for 7 reveals that 509 is not divisible by 7.

Since 509 is not divisible by any numbers other than 1 and 509, it is a prime number.

The prime number chart is a tool created by using a method called “The Sieve of Eratosthenes.” In this method, we follow the following steps.

Step 1: Write numbers sequentially up to a larger range, such as 1 to 1000.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Start marking prime numbers like 2, 3, 5, etc., and cross out their multiples.

Step 4: Continue this process until you reach the largest number in your range. Through this process, prime numbers are identified, and 509 is found in the list of prime numbers, confirming it is prime.

Prime factorization is a process of breaking down a number into prime factors. Then multiply those factors to obtain the original number.

Step 1: Attempt to write 509 as a product of two numbers.

Step 2: Try dividing 509 by smaller prime numbers (2, 3, 5, 7, etc.).

Step 3: By testing, we find that 509 is not divisible by any primes up to its approximate square root (~22.5).

Step 4: As no prime factors are found other than 509 itself, 509 remains undivided, confirming it is a prime number.

• Prime numbers: Natural numbers greater than 1 with only two divisors, 1 and itself.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers.

• Divisibility: A number's ability to be divided by another number without leaving a remainder.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.

• Square root: A value that, when multiplied by itself, gives the original number.